Do VR, AR and MR use any machine learning or deep learning?
TL;DR

Yes for all the 3 cases 

Details

VR

Even if the environment is virtual there is another thing to perceive which or better who is real: the user 

Deep Learning can be used for things like: eye tracking, gesture recognition, voice recognition, ... 

Furthermore, in games,  you could think about more complex applications like understanding the way the user plays the game and react accordingly 

AR and MR

In addition to what has been said for VR, in these cases you also have to integrate the environment in the final user experience so you ultimately have to perceive it and there multiple specific applications: object detection, planes detection, depth estimation, Visual Odometry (e.g. as feature detection module), SLAM, ... 

These days Deep Learning is achieving Starte-Of-The-Art (SOTA) in many different tasks far beyond the classification and detection, but one things is research so published papers and so on and another thing is a product 
so to sum up every VR, MR, AR vendor is probably relying on an hybrid solution where traditional algorithms and deep learning coexist but so far it seems the trend is deep learning is eating the traditional algos up day after day 

References

The rising force of deep learning in VR and AR

Virtual Reality And Machine Learning Go Hand In Hand